Does synthesis support effect HQ rate?

Yes! Synthesis support modifies your effective skill level. It's generally accepted that when you reach a certain level of skill above the cap of a recipe, you get more chances to HQ. So, if you are 28 levels above a recipe, and get advanced support, your effective skill is 31 levels above the recipe, and you move up an HQ tier and get more chances to HQ. Note that support does not effect your skillup chance, as that is determined by your real skill level.